SSRN-id1525762

Riding Bubbles

Nadja Guenster, Erik Kole, and Ben Jacobsen



ERIM REPORT SERIES RESEARCH IN MANAGEMENT

ERIM Report Series reference number Publication Number of pages Persistent paper URL Email address corresponding author Address ERS-2009-058-F&A December 2009 80 http://hdl.handle.net/1765/17525 kole@ese.eur.nl Erasmus Research Institute of Management (ERIM) RSM Erasmus University / Erasmus School of Economics Erasmus Universiteit Rotterdam P.O.Box 1738 3000 DR Rotterdam, The Netherlands Phone: Fax: Email: Internet: + 31 10 408 1182 + 31 10 408 9640 info@erim.eur.nl www.erim.eur.nl



Bibliographic data and classifications of all the ERIM reports are also available on the ERIM website: www.erim.eur.nl



ERASMUS RESEARCH INSTITUTE OF MANAGEMENT REPORT SERIES RESEARCH IN MANAGEMENT ABSTRACT AND KEYWORDS

Abstract We empirically analyze rational investors' optimal response to asset price bubbles. We define bubbles as a sudden acceleration of price growth beyond the growth in fundamental value given by an asset pricing model. Our new bubble detection method requires only a limited time-series of historical returns. We apply our method to US industries and find strong statistical and economic support for the riding bubbles hypothesis: when an investor detects a bubble, her optimal portfolio weight increases significantly. A dynamic riding bubble strategy that uses only real-time information earns abnormal annual returns of 3% to 8%. Free Keywords Availability bubbles, limits to arbitrage, market efficiency, structural breaks The ERIM Report Series is distributed through the following platforms: Academic Repository at Erasmus University (DEAR), DEAR ERIM Series Portal Social Science Research Network (SSRN), SSRN ERIM Series Webpage Research Papers in Economics (REPEC), REPEC ERIM Series Webpage Classifications The electronic versions of the papers in the ERIM report Series contain bibliographic metadata by the following classification systems: Library of Congress Classification, (LCC) LCC Webpage Journal of Economic Literature, (JEL), JEL Webpage ACM Computing Classification System CCS Webpage Inspec Classification scheme (ICS), ICS Webpage



Riding Bubbles∗

Nadja Guenster†1 , Erik Kole2 , and Ben Jacobsen3

2 Econometric



Department, Maastricht University, The Netherlands Institute, Erasmus School of Economics, Erasmus University Rotterdam, The Netherlands 3 Dept. of Commerce, Massey University, Auckland, New Zealand



1 Finance



December 9, 2009



Abstract We empirically analyze rational investors’ optimal response to asset price bubbles. We define bubbles as a sudden acceleration of price growth beyond the growth in fundamental value given by an asset pricing model. Our new bubble detection method requires only a limited time-series of historical returns. We apply our method to US industries and find strong statistical and economic support for the riding bubbles hypothesis: when an investor detects a bubble, her optimal portfolio weight increases significantly. A dynamic riding bubble strategy that uses only real-time information earns abnormal annual returns of 3% to 8%. Key words: bubbles, limits to arbitrage, market efficiency, structural breaks JEL classification: G10, G14, C14



We thank Martin Bohl, Markus Brunnermeier, Phil Davies, Jeroen Derwall, Dick van Dijk, Mathijs van Dijk, Ingolf Dittmann, Alex Edmans, Piet Eichholtz, R¨diger Fahlenbrach, Robert Flood, Campbell u Harvey, Kewei Hou, Raymond Kan, Andrew Karolyi, Kees Koedijk, Philipp Koziol, Marcel Marekwica, ˇ Stefan Nagel, Luboˇ P´stor, Thomas Post, Adriano Rampini, Lev Ratnovski, Erwan Le Saout, Peter Schots a man, Sophie Shive, Marta Szymanowska, Rene Stulz, Jerome Taillard, Marno Verbeek, and participants at the 2009 European Finance Association (EFA) Annual Meeting, 2009 McGill Global Asset Management Conference, the 2009 European Winter Finance Conference, the 2008 Society for Financial Econometrics Conference, the 2008 Washington Area Finance Association Meeting, the 2008 Norges Bank workshop in Venastul (Norway), WHU’s Campus for Finance 2008 conference, the 2007 French Finance Association Meeting, the 2007 CEPR-Banque de France conference, Erasmus University, Maastricht University and at the PhD Seminar Series at Ohio State University for helpful comments and discussions. We thank Sandra Sizer for editorial assistance. Kole gratefully acknowledges financial support from the Vereniging Trustfonds Erasmus Universiteit Rotterdam. E-mail addresses: nk.guenster@maastrichtuniversity.nl, kole@ese.eur.nl, b.jacobsen@massey.ac.nz. † Corresponding author: Tongersestraat 53, 6200 MD Maastricht, The Netherlands, Tel. +31 43 388 4947.



∗



1



Introduction

Should investors who suspect an asset price bubble trade against it or with it? Or, should they just step aside and wait until prices get back to normal? Different strands of the theoretical literature support all three, albeit contradictory, propositions. First, the efficient market hypothesis predicts that rational investors trade against bubbles by shorting the overvalued asset. Rational investors will “cause these “bubbles” to burst”(see Fama, 1965, p. 38) before they can even emerge. Second, in stark contrast, De Long, Shleifer, Summers, and Waldmann (1990b) and Abreu and Brunnermeier (2003) propose that when investors detect a bubble, they should actively increase their holdings, i.e., “ride the bubble”. In these models, rational investors actively fuel bubbles. Third, the limits to arbitrage literature, for example De Long, Shleifer, Summers, and Waldmann (1990a), Dow and Gorton (1994) and Shleifer and Vishny (1997), posits that it neither shorting nor riding the bubble is optimal. Instead, investors should limit their positions if they suspect a bubble (hereafter: “sideline”). The empirical literature does not focus on investor’s optimal strategy. Instead, empirical studies (e.g., Brunnermeier and Nagel, 2004; Griffin, Harris, and Topaloglu, 2006; Temin and Voth, 2004; Greenwood and Nagel, 2009; Dass, Massa, and Patgiri, 2008) describe the trading strategies of different investors during historical bubbles. These studies show that investors pursued a variety of strategies, depending on their incentives, experience, and type. In this paper, we empirically analyze the optimal strategy of a rational investor who learns about a bubble. In contrast to other empirical studies, we take a normative perspective. We systematically identify bubbles and analyze the optimal strategy for a broad range of rational investors. Therefore, our findings are neither limited to a specific type of investor nor to a specific bubble period. This normative, systematic approach enables us to distinguish between the competing theoretical propositions and sheds light on the mixed empirical evidence. Bubbles are caused by fashions and fads, usually in combination with technological changes about which investors are “irrationally exuberant”(Shiller, 2000). Because firms in one industry are exposed to the same changes in business environment and technology,



2



bubbles tend to start in specific industries. Examples are the railway boom, the electricity boom, the tronics boom or the recent internet bubble. Therefore, we consider industry portfolios as intuitively appealing assets for our analysis, and use the Fama and French (1997) sample of 48 US industry returns. Central to our approach is a new bubble identification method. This method relies on two main characteristics of asset price bubbles, described for example by Abreu and Brunnermeier (2003): (1) the growth rate of the price is higher than the growth rate of fundamental value and (2) the growth rate of the price experiences a sudden acceleration consistent with the features of the Minsky model described by Kindleberger (2000). An investor concludes that a bubble exists if both conditions are fulfilled. A bubble ends when the investor observed a crash during the previous six months or when a bubble is no longer detectable. The investor estimates the growth rate of fundamental value based on one of three different asset pricing models, the Capital Asset Pricing Model (CAPM), the Fama-French (1993) Three-Factor Model (3F-Model), and the Carhart (1997) Four-Factor Model (4F-Model). To detect a sudden acceleration, the investor conducts a structural change test as in Andrews (1993). This bubble identification method is explicitly designed to answer our research question. Since we want to use only a short time-series of data, we do not use the cointegration approach of Campbell and Shiller (1987) to identify bubbles. Just like a “real” investor, we cannot identify bubbles without error. At times, we might miss a bubble or identify a bubble that is truly something else. This feature makes our method realistic by introducing the noise and uncertainty real-world investors must deal with. Our bubbles are clearly distinct from the rational bubbles described, for example, by Blanchard and Watson (1982). By definition, investors are indifferent to selling or investing in a rational bubble, so these bubbles are irrelevant to our research question. We compare the return distribution following the detection of a bubble to the distribution if no bubble was detected. Our results show that investors confront a substantially different risk-return trade-off following bubbles. First, the abnormal returns following bubbles are, on average, significantly higher than if the investor had not detected a bubble. Second, we find significant increases for various risk measures. The difference in monthly returns is about 0.34% for the 4F-Model and 0.39% for the 3F-Model. For the CAPM3



based abnormal returns, it is 0.77% per month. Volatility shows a relative increase of about 14% for the CAPM and 3F-Model and 18% for abnormal returns based on the 4F-Model. Downside risk measures like Value-at-Risk and Expected Shortfall show comparable increases. To find out whether an investor should ride, sideline, or short the bubble, we investigate the asset allocation implications of our findings. An investor with a power-utility function substantially increases her optimal weight in an asset if she learns about a bubble. This increase is statistically and economically significant with portfolio weights (as a fraction of wealth) going up by 0.64 (4F-Model), 0.91 (3F-Model) and 1.61 (CAPM). We also compute the risk-free return the investor would require as compensation for not increasing the weight when she detects a bubble. On a yearly basis, these certainty equivalents add up to 1.12% (4F-Model), 2.11% (3F-Model) and a stunning 7.04% (CAPM). We examine the robustness of our conclusions to different preferences. Investors who are particularly averse to skewness, kurtosis or downside risk averse investors would also ride bubbles. A simple dynamic investment strategy, for which we use only real-time information, earns annual abnormal returns of about 3% to 4% based on the 3F-and 4F-Models, and 8% for the CAPM. Riding bubbles is an attractive strategy for investors who have information only on past returns. Our results show that it is not optimal for rational arbitrageurs to exert a correcting force on prices during bubbles. Instead, our empirical findings confirm the theoretical predictions made by Abreu and Brunnermeier (2003) and De Long, Shleifer, Summers, and Waldmann (1990b) that it is optimal for rational investors to ride bubbles and thereby fuel them. Therefore, markets will not resolve bubbles by themselves. Our findings contribute to explaining the significance of bubbles we have experienced over the last decade. We examine whether alternative theories can explain our findings. One obvious alternative is a misspecification of the asset pricing model. We show that an omitted structural break in any of the risk factors, an omitted risk factor or a structural break in an omitted risk factor cannot explain our results. We compare the industry bubbles to industry momentum, as described by Moskowitz and Grinblatt (1999), and find that they are distinct. A sequence of good news to which the market potentially underreacts is an unlikely alter4



native explanation because the bubbles are followed by negative returns over the following two years. We perform an extensive examination of the robustness of our results to choices made in the analysis. We confirm that our findings are neither driven by specific industries nor by a small subset of bubble observations. The paper is organized as follows. First, we give a short introduction to the literature, Second, we describe our data set. In the third section, we explain how we identify bubbles. We analyze the relation between the detection of a bubble and future abnormal returns in section 4. In section 5, we analyze how a power-utility investor would allocate her portfolio based on our findings. Section 6 discusses a real-time strategy and its results. In section 7, we investigate whether alternative theories can explain our findings. Section 8 examines the robustness of our results. Section 9 concludes.



1



Asset Price Bubbles and Investment Strategies



Theoretical studies have yet to reach consensus on a rational investor’s optimal response to asset price bubbles. From an efficient market perspective, arbitrage by sophisticated traders precludes the existence of bubbles. Fama (1965, p.38) states “If there are many sophisticated traders in the market, however, they may cause these bubbles to burst before they have a chance to really get under way”. Despite the fact that theoretically, bubbles can be ruled out, we observe them. Therefore, a second line of research explains why arbitrageurs do not trade against mispricing. The central idea, introduced by De Long, Shleifer, Summers, and Waldmann (1990a), is that noise traders can cause prices to diverge even further from fundamental value. The risk that the price of the asset will not return to its fundamental value, or that the gap widens, outweighs the potential gain from arbitrage trading. De Long, Shleifer, Summers, and Waldmann (1990a) show that even in the absence of any fundamental risk, it is optimal for risk-averse arbitrageurs with finite horizons to refrain from trading against the mispricing. For assets with fundamental risk, such as equity, not even arbitrageurs with long-term strategies would be able to certainly profit from trading against the mispricing. Several authors have extended De Long, Shleifer, Summers, and Waldmann (1990a)’s analysis. Dow and Gorton (1994) introduce transaction costs and show that it is only 5



profitable for an informed trader to act on her information if she can be sure that subsequent arbitrageurs also trade in the same direction, i.e. if arbitrage chains are unbroken. Shleifer and Vishny (1997) show that arbitrageurs who trade on behalf of uninformed clients face capital constraints and limited horizons. Therefore, if there is noise-trader risk, they do not trade against mispricing. Goldman and Slezak (2003) arrive at similar conclusions for mutual fund managers, who inherit portfolios and manage them for a limited time period. A third line of theoretical research, introduced by De Long, Shleifer, Summers, and Waldmann (1990b) predicts that rational arbitrageurs actively invest in the asset price bubble. These arbitrageurs not only refrain from trading against the bubble, they also drive prices further away from fundamental value. Abreu and Brunnermeier (2003) develop a model showing that arbitrageurs have an incentive to ride bubbles if there is synchronization risk. If each individual arbitrageur is not able to burst the bubble on her own and concludes that other arbitrageurs are unlikely to attack it, then her optimal choice is to profit from the noise traders by riding the bubble. Several empirical papers provide, directly or indirectly, evidence either for or against the different theoretical propositions. Temin and Voth (2004) document that a very sophisticated investor, Hoare’s Bank, was well aware of the South Sea Bubble of 1720. The bank actively invested in, and profited from, the bubble. Brunnermeier and Nagel (2004) show that hedge funds were actively investing in tech stocks during the recent internet bubble. Furthermore, hedge fund managers were able to time the crash and therefore profited substantially from riding the bubble. These two papers show that two highly sophisticated types of investors were following the strategy proposed in Abreu and Brunnermeier (2003) during the two historical bubble episodes. Griffin, Harris, and Topaloglu (2006) analyze the trading behavior of different types of investors during the tech bubble by comparing the direction of the trade to contemporaneous and lagged returns. These authors classify investors according to the brokerage house through which the trade is executed. Their findings suggest that institutional investors, hedge funds, and day traders were fueling the bubble and then causing it to burst. These groups were trading in the same direction as contemporaneous market movements. However, hedge funds and day traders traded contrarian relative to lagged market returns, indicating that they were not consistently riding the bubble. The trading behavior of other 6



individual investors and of derivative traders was, overall, more contrarian. Based on the results from Griffin, Harris, and Topaloglu (2006), it seems that some investors were riding the bubble while others were trading against it. A few studies analyze the behavior of mutual fund managers during the internet bubble. Although these studies do not directly aim at providing evidence either in favor of or against the different hypotheses outlined above, they still provide interesting insights. Greenwood and Nagel (2009) document that as long as the tech market was rising, younger mutual fund managers were overweighing technology stocks compared to their benchmark. Older mutual fund managers did not exhibit this trend-chasing behavior. However, unlike hedge fund managers, younger mutual fund managers were not successful at predicting the downturn. Dass, Massa, and Patgiri (2008) analyze the impact of incentive contracts on mutual fund managers’ investment decisions during the tech bubble. Advisory contracts with high performance-based incentives induced managers to diverge from the herd and not ride the bubble. The prospect of receiving a high pay-off by being the best outweighs the potential reputation loss of being worse than the average. In line with this result, Dass, Massa, and Patgiri (2008) also find that managers with low incentive contracts outperformed the managers with high incentive contracts during the bubble. However, it again seems that the mutual fund managers were not able to time the burst of the bubble. In the period 2001-2003, the low-incentive managers substantially underperformed the high-incentive managers. Overall, mutual fund managers’ investment strategies were more heterogenous than the strategies of hedge fund managers: certain groups were riding the bubble, while others did not. The mutual fund managers who were riding the bubble profited from it as long as the bubble lasted, but they incurred huge losses when the bubble burst. The empirical studies provide mixed evidence on the behavior of different types of investors. Based on Griffin, Harris, and Topaloglu (2006), it seems that institutional investors were riding the bubble, but individual investors, except day traders, refrained. However, Greenwood and Nagel (2009) and Dass, Massa, and Patgiri (2008) show that some mutual funds were riding the bubble while others were not. Brunnermeier and Nagel (2004) find that hedge funds consistently increased their holdings until the peak of the bubble, while Griffin, Harris, and Topaloglu (2006) provide evidence that hedge fund managers traded contrarian relative to the previous day’s return. 7



The different empirical studies describe the trading strategies of specific investor types during historical bubble episodes. The aim of this study is to reach a general conclusion that is not limited to a specific investor type or a specific bubble period. In contrast to previous studies, we do not take a descriptive but a normative perspective, so we would like to know an investor’s optimal response. By doing so, we can distinguish among the competing theoretical propositions and contribute to a better understanding of the existence and persistence of bubbles.



2



Data Description



Throughout our analysis, we use the 48 industry indexes constructed by Fama and French (1997) and available on Ken French’s website.1 We use the industry returns constructed with the new specifications. The data set consists of monthly value-weighted returns from July 1926 to December 2006. Value-weighted returns reflect the portfolio returns earned by real-world investors and ensure that small-cap stocks do not dominate our findings. In the CAPM estimation, we measure the risk-free rate and the market index by the one-month Treasury bill rate from Ibbotson Associates and the CRSP All Share Index, respectively. For the Fama and French (1993) model we also include the factor portfolios “High-Minus-Low”(HML) and “Small-Minus-Big” (SMB). In the Carhart (1997) model we add “Momentum” (MOM) as a fourth factor. We also obtain the factor portfolios and the risk-free rate from Ken French’s website. Since these data are already used in other studies and freely available, we do not further elaborate on the data set here. However, for the interested reader, Table A.1, in Appendix A, provides descriptive statistics on the industry returns.



3



Defining Bubbles



“The word “bubble” is widely used to mean very different things...” (Cochrane, 2001). Although the exact specification of asset bubbles differs among theoretical models, two

1



See http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.



8



prominent characteristics of bubbles are common to many models and to anecdotal descriptions. First, as far back as the origins of the rational bubble literature (see Blanchard and Watson, 1982), a distinctive characteristic of bubbles is that the price grows faster than the fundamental value of the asset. Second, a bubble represents a sudden acceleration in the growth rate of the price. Abreu and Brunnermeier (2003) use these two characteristics to define bubbles. Kindleberger (2000, p. 14) characterizes a bubble by a displacement in a Minsky model and Shiller (2000) relates bubbles to “new economy thinking”. In the theoretical models, an investor either receives a signal that the asset is experiencing a bubble or she has known this fact all along (see e.g., Abreu and Brunnermeier, 2003; De Long, Shleifer, Summers, and Waldmann, 1990b). Since our purpose is to investigate the different theoretical predictions in a real-world setting, we require that the investor infers bubbles from publicly available information. Our conclusions must be generalizable to many investors rather than limited to a few very sophisticated arbitrageurs who can afford to spend a substantial amount of time and money on acquiring a superior information set. Therefore, we restrict the information set to past prices. Without a long time-series of information, which potentially includes the burst of the bubble, an investor cannot identify bubbles with certainty. Sometimes she might miss a bubble and at other times she mistakenly believes that there is a bubble. Our realtime bubble identification method introduces noise into our analysis and thus weakens our findings. However, we believe that this approach is more realistic, because investors face similar uncertainties. For example, there is an ongoing debate on whether the Tulip Mania of the 1600s and the internet bubble were real bubbles, or whether the high prices were justified by fundamentals (see for example Garber, 1989; Ofek and Richardson, 2002; P´stor a and Veronesi, 2006). This discussion illustrates that no one, even with 20/20 hindsight, can identify bubbles with certainty. Our bubble definition is designed to capture the two basic characteristics of bubbles outlined in the bubble literature while only using past price information. To adjust returns for the growth rate of fundamental value, we use three different asset pricing models: the CAPM, the Fama and French (1993) Model (3F-Model) or the Carhart (1997) Model (4FModel). To identify a sudden acceleration in price growth, we test for a positive structural break in returns, which is not explained by the asset pricing models. In addition, to capture 9



the idea that the asset price grows faster than the fundamental value, we require significantly positive anomalous returns following the break. Unlike the cointegration approach put forward by Campbell and Shiller (1987) or the regime-switching model proposed by Brooks and Katsaris (2005), our bubble identification method makes it possible to use a limited history of past price information, at a level that we believe is also available to investors. We also do not need to make any assumption on investors’ expectations of future earnings. Formally, we investigate whether an asset experiences a bubble at time t by estimating: rτ = ατ + β ′ fτ + ετ , E[ετ ] = 0, E[ε2 ] = σ 2 , τ τ = t − T + 1, . . . , t (1)



where rτ is the asset’s excess return and T is the estimation window, which typically equals 120 months. The vector fτ represents the set of risk factors. In our first specification, we estimate the CAPM developed by Sharpe (1964) and Lintner (1965). Second, we augment the CAPM by the factors SMB and HML resulting in the 3F-Model. Third, we estimate the 4F-Model, which also includes the momentum factor. Our test procedure concentrates on ατ . To capture the two basic characteristics of a bubble, we interpret a bubble as a structural break in ατ , after which ατ is significantly positive. Our setup closely follows the structural break literature (see Andrews, 1993; Hansen, 2001), and the null hypothesis of no bubbles implies that ατ does not change significantly during the test period: H0 : ατ = α0 for all τ. (2)



The alternative is that we observe a structural break in ατ . Since we have no a priori expectations of when a bubble starts, we test for different breakpoints. Because we are interested in recent bubbles, we require that the bubble lasts until time t. In addition, we require a bubble to be a prolonged acceleration in price growth and set its minimum length to 12 months. Its maximum length is five years. Formally, the alternative hypothesis reads:   α1 (ζ) for τ = t − T + 1, . . . , t − ζ H1T (ζ) : ατ = (3)  α2 (ζ) for τ = t − ζ + 1, . . . , t, with α2 (ζ) > α1 (ζ), where ζ ranges from 12 to 60, α1 (ζ) refers to the first part of our test period, and α2 (ζ) to the second part. For each value of ζ we calculate the t-statistic for 10



the hypothesis α1 (ζ) = α2 (ζ). We select the breakpoint ζ with the largest test statistic and determine its critical value based on the tables in Andrews (1993).2 If we reject H0 in favor of H1T , then we subsequently test whether α2 (ζ) is significantly larger than zero. Based on Bai (1994) and Bai and Perron (1998), who derive the asymptotic distribution of estimates in a model with structural change, we apply a standard t-test. If both criteria are fulfilled, then we conclude that the asset is experiencing a bubble. We use low (95%), medium (97.5%), and high (99%) confidence levels to determine the critical values. We consider the medium level as the base case scenario. Every time we detect a bubble, we count the number of months since inception and thus define the length of the bubble. In addition, we define the strength of the bubble as the t-statistic of α2 (ζ). In many models, ranging from Blanchard and Watson (1982) to Abreu and Brunnermeier (2003), bubbles end with a crash. Therefore, we take a crash in the innovations ετ during the last κ months as a signal that the bubble has ended. We define a crash as a value of ετ below a threshold, a multiple k of the standard deviation σ. Typically we take κ equal to six months, and the threshold multiple equal to -2. If bubbles do not end with a crash, but instead deflate, then the estimate for α2 will decrease over time. If the deflation is strong enough, then α2 will not pass the second test. In that case, the investor can no longer detect the bubble. Figure 1(a) shows the number of bubbles per industry, which we compute as the unbroken sequence of months a bubble is detected. On average, each industry experiences about two bubbles. We find slightly fewer bubbles for the 3F- and 4F-Models than for the CAPM. Figure 1(b) shows the distribution of bubble detections across industries. Every time we identify a bubble we count it as a detection. Thus, we can detect “one” bubble many times. Both the number of bubbles and the number of bubble detections are spread widely across industries, so we conclude that our findings are not be driven by a specific bubble or industry.

2



Andrews (1993) actually discusses two-sided tests for the detection of a structural break, whereas our



alternative hypothesis is one-sided. Estrella and Rodriguez (2005) derive the corresponding asymptotic distribution of the t-statistic. They show in their Figure 1 that the common approach of halving the p-value for a given critical value when moving from a two-sided to a one-sided alternative gives a good approximation. Therefore, we use the tables provided by Andrews (1993).



11



[Figure 1 about here.] To investigate whether our detection method discovers economically meaningful bubbles, we compute standardized abnormal returns as well as “raw” returns during bubbles. For the abnormal returns, we use the same factor models as in the bubble identification. We estimate this model over the previous 120 months to compute the abnormal return for the following month according to: ˆ ηt+1 = rt+1 − β ′ ft+1 , (4)



ˆ where the rt+1 is the excess return at t + 1, and β ′ is a vector of estimates based on the regression in Equation (1) under the null hypothesis. To accommodate time-varying volatilities and different volatilities across industries, we standardize the abnormal returns by dividing them by the residual standard deviation: ηi,t ≡ ηi,t /σi,t . Tables A.2 to A.4 in Appendix A provide summary statistics of the stan˜ dardized abnormal returns per industry. On average, the standardized abnormal returns are close to zero. The volatility differs from one in statistical significance, but the deviations are economically small, ranging from 0.04 for the CAPM to 0.08 for the 4F-Model. Although the average abnormal returns for the complete sample are close to zero, Table 1 shows that the raw and abnormal returns during bubbles are economically large. The standardized abnormal returns during bubbles for the different models are 0.47, 0.49, and 0.51 per month for the CAPM, the 3F-Model and the 4F-Model, respectively. We note that the average idiosyncratic volatility ranges from 4.14% for the CAPM to 3.83% for the 3F-Model and 3.79% for the 4F-Model. Thus, if we assume an average idiosyncratic return volatility of 4%, these returns translate into annual abnormal returns of about 23%. We also find that the residual volatility of the standardized abnormal returns is substantially larger than one. These findings indicate that we observe economically meaningful deviations from the null hypothesis. [Table 1 about here.] Although our bubble identification method is inherently based on abnormal returns, for illustrative purposes we compute the raw returns during bubbles. The raw return of 12



the average industry during a bubble is 30.6% per year when we apply the CAPM or the 4F-Model, and 32.4% when we use the 3F-Model. In line with our findings for the returns, we also find that α1 is indistinguishable from zero while α2 is large and positive. We continue by examining the length of the bubbles. To identify the returns that belong to one bubble, we mark all months that ever show up in a bubble detection for each industry. We count a series of detections without an interruption as one bubble. For all different asset pricing models we find that the mean and median bubble length is about 30 months. This finding shows that the bubbles we identify are a long-term phenomenon and do not vanish quickly after their detection. Appendix A, Tables A.5 to A.7 provide bubble statistics per industry.



4



The Risks and Returns After Bubbles



To examine the profitability of riding bubbles, we analyze the risk-return trade-off following the detection of a bubble. Given that the investor has detected a bubble up to month t, we compare the abnormal returns in month t + 1 to the abnormal returns if no bubble has been detected. Thus, we base this comparison solely on post-identification returns. It does not include observations used in the identification of the bubble. Table 2, Panel A shows the characteristics of return distribution if the bubble detection and abnormal return construction is based on the CAPM. Panels B and C show the results for the 3F- and 4F-Models. We use 10.000 bootstraps to construct standard errors and conduct tests. Throughout all specifications, we observe significantly positive abnormal returns after the detection of a bubble. The returns are also economically large. The mean abnormal return is 0.77% (0.186 × 4.14%) per month if we base our estimates on the CAPM. For the 3F- and 4F-Models, it is 0.39 (0.101 × 3.83%) and 0.34% (0.089 × 3.79%), respectively. The estimates of median abnormal returns are somewhat smaller ranging from 0.58% (0.139 × 4.14%) per month for the CAPM to 0.23% (0.060 × 3.79%) per month for the 3F-Model. If we do not detect a bubble, which is the case for most of our sample period, then the abnormal returns are either close to zero or slightly negative. Again, we observe that the median is smaller than the average. The p-values in the final column of each panel indicate that, depending on the presence or absence of a bubble, the differences 13



in abnormal returns are significant below the 1% significance level for all asset pricing models.3 On an annual basis, the return differentials range from 3.7% for the 4F-Model to 8.8% for the CAPM. To ensure that our findings are not driven by a specific bubble or industry, Table A.8 in Appendix A shows the abnormal returns per industry for each of the factor models. [Table 2 about here.] The evidence so far supports the idea that riding bubbles is a highly profitable strategy. However, the different risk measures presented in Table 2 indicate that it is also a risky strategy. For all three asset pricing models, we find that the abnormal return volatility following a bubble is significantly larger than it would have been if no bubble had been detected. Furthermore, especially for the abnormal returns based on the 3F- and the 4FModels, we observe significant differences in downside risk. For example, multiplying again by the idiosyncratic volatilities, the Value-at-Risk (Expected Shortfall) at the 95% level based on the 4F-Model is 7.31% (9.94%) if we detect a bubble compared to 6.28% (8.65%) if there is no bubble. When we compute abnormal returns based on the CAPM, in contrast to Chen, Hong, and Stein (2001), we observe that skewness is larger after the detection of a bubble. Once we account for size and style (Panel B), the relation diminishes and there is no significant difference any more. If we also account for momentum, in Panel C, the relation reverses. Again, however, the difference is not statistically significant. It seems that kurtosis is lower when we detect a bubble than in the absence of a bubble, suggesting that the tails of the distribution are fatter if no bubble is detected. However, one has to take into account the different number of observations: We have much more observations without than with a bubble detection. Therefore, we also have more extreme observations if we did not detect a bubble. The following figures, which are based on kernel estimation, adjust for this difference and show that the tails are actually fatter after a bubble.

3



We test for the equality of the different moments under the null hypothesis of equal distributions if a



bubble was detected or no bubble was detected. We have also conducted the tests under the null hypothesis of equal moments. Since the statistical significance of our findings across the different formulations of the null hypothesis is very similar, these results are available upon request from the authors.



14



To gain a clearer impression of the complete distribution of abnormal returns, Figures 2(a), 2(c) and 2(e) plot the distribution of abnormal returns based on the three different asset pricing models with and without bubble detection. An investor who is riding bubbles not only confronts the upside potential of positive abnormal returns, but also the downside risk of extremely negative returns. In line with the positive abnormal returns we document above, the distribution after the detection of a bubble is shifted to the right. However, especially for two standard deviations below zero and lower, we observe that the return distributions after bubbles have a prominent left tail (see also Figures 2(b), 2(d), and 2(f)). These relatively rare but extremely negative returns suggest that bubbles may deflate quickly in crashes. [Figure 2 about here.] The evidence of large positive abnormal returns shows that investors have a strong incentive to ride bubbles. However, because returns are also riskier after the detection of a bubble, it is not clear how identifying a bubble affects a rational investor’s asset allocation.



5



The Asset Allocation Decision



We investigate how the detection of a bubble affects an investor’s optimal asset allocation. If the investor substantially increases her weight in the asset when she learns about a bubble, we conclude that it is optimal to ride bubbles. To the contrary, if the investor substantially decreases her position in an asset following bubble detection, we conclude that shorting is optimal. Given the higher positive abnormal returns following a bubble than if no bubble is detected, shorting cannot be optimal. The third alternative is that the higher risk after bubble detection outweighs the increase in average abnormal returns. In this case, a risk-averse investor might refrain from increasing her allocation to the asset bubble and we conclude that she is sidelining. The investor can allocate her wealth W at t between a single risky asset, which is represented by the typical industry, and a risk-free asset. We assume that the investor can hedge systematic risk factors in industry returns. This approach makes it possible for us to isolate the impact of the bubble on the optimal asset allocation from systematic 15



determinants of industry returns. The risky asset earns a return ri and the return on the risk-free asset is denoted as rf . The portfolio return rp obeys:

′ rp = wri − wβi f + (1 − w)rf = rf + wηi



(5)



where w is the fraction of wealth invested in the risky asset and ηi is the abnormal return of the risky asset. The investor’s preferences over wealth are described by a power-utility function: U (W ) = W 1−γ − 1 1−γ γ ̸= 1, W > 0 (6)



where γ is the coefficient of risk aversion. For W ≤ 0, the investor is bankrupt and we set utility to −∞. In our main analysis, we focus on a power-utility investor. However, to ensure the robustness of our conclusions, we repeat the analysis for investors who have different skewness and kurtosis preferences, mean-variance investors and downside-risk averse investors. These robustness checks confirm our conclusions. Based on Equation (5), the next period’s wealth is given by Wt+1 = Wt (1 + rf + wηi ) and the investor’s problem is to solve: [( )(1−γ) ] Wt (1 + rf + wηi ) max Et w 1−γ which results in the first-order condition [ ] E (1 + rf + w∗ ηi )−γ ηi = 0. (8)



(7)



We do not make assumptions on the distribution of abnormal returns η and instead use its empirical distribution. Therefore, the solution of Equation (8) is based on numerical approximations. We also compare the expected utility in the presence and absence of a bubble, i.e. VB ≡ max E[U (1 + rF + wηi )|IB ] and VNB ≡ max E[U (1 + rf + wηi )|INB ]. We use I to denote the information set, with IB (INB ) indicating that a bubble has (has not) been detected. If riding bubbles is the optimal strategy, then the expected utility should be significantly higher upon bubble detection than if no bubble is detected. If sideling is optimal, then the expected utility of both scenarios should be about equal. 16



To determine the economic significance of trading on bubble information, we compute the certainty equivalent return. We calculate the risk-free return that the investor would require as a compensation for not changing her portfolio when she detects a bubble. The optimal weight with(out) prior bubble detection is denoted as wB (wNB ). The certainty equivalent return λ satisfies the condition E[U (1 + rf + wB ηi )|IB ] = E[U (1 + rf + wNB ηi + λ)|IB ] and we solve for λ numerically. Table 3 presents the results. We put the idiosyncratic volatility of the industry equal to the pooled averages of the respective models, and we use the long-run average of the risk-free rate of 30.5 basis points per month (3.67% per year). Columns two to four show that when an investor detects a bubble, the optimal weight assigned to the risky asset increases substantially. For an investor with a moderate risk aversion level of two, the optimal weight increases from 0.1 to 1.71 for the abnormal returns based on the CAPM. For the 3F-Model and 4F-Model the optimal weight rises from zero to 0.91, and from 0.11 to 0.74, respectively.4 The standard errors (shown in parentheses) indicate that the weight allocated to the risky asset in the absence of a bubble is indistinguishable from zero. When the investor detects a bubble, it becomes statistically significantly different from zero. The p-values in column 4 show that the difference in weight is statistically significant at the 5% level for all specifications. Again, we conduct the test under the null hypothesis that the abnormal return distributions are equal in the presence and absence of a bubble. Because the utility function takes the complete distribution into account, this null-hypothesis exactly represents the relevance (irrelevance) of information on bubbles. [Table 3 about here.] We also investigate how the change in optimal weight is affected by different riskaversion levels. Even for rather high levels of risk aversion, the difference in portfolio

4



(9)



In the CAPM-case, the investor takes a levered position, which can theoretically lead to negative



wealth with a non-zero probability. We find a levered position to be optimal, because the optimization is based on the empirical distribution. Including the possibility of extreme negative returns would lead to a maximum investment of one. For higher degrees of risk aversion, allocations would not change.



17



weights is economically significant. Even for a risk aversion level of ten, our most conservative estimate, which we base on the 4F-Model, indicates an increase in the optimal weight of 13%. Using the approximation E[(1 + rf + w∗ ηi )−γ ηi ] ≈ E[exp (−γ(rf + w∗ ηi ))ηi ] for the first-order condition in Equation (8) shows that a multiplication of γ with a constant requires a division of w∗ with the same factor to maintain optimality. It also explains why the p-values depend very little on the level of risk aversion. To test wB = wNB , we construct a series of portfolio differences, based on a bootstrap. Since both weights are approximately inversely related to risk aversion, the same holds for the difference, and the test is only marginally affected. In line with our results for the optimal weight, we find that riding bubbles is associated with significant increases in expected utility. For an investor with a risk-aversion of γ = 2, the expected utility rises from typical values around 3.06 · 10−3 to 4.29 · 10−3 for the 4FModel and 9.55 · 10−3 for the CAPM. Investors require a sizable compensation for not updating their portfolio when they detect bubbles. For the CAPM, the investor would require a risk-free return of 7% annually. The certainty equivalent returns for the 3F-Model and 4F-Model are 2.11% and 1.12% for an investor with a risk-aversion level of two. As risk aversion decreases, the certainty equivalent increases. The certainty equivalent is significantly positive across all the different specifications, as shown by the p-values in Column 9 of Table 3. Our results show strong statistical and economic support for the riding bubbles hypothesis. Increases in portfolio weights are large, and the certainty equivalent returns for a bubble riding strategy are non-trivial. The increases in portfolio weights are sometimes extreme, with investors choosing positions exceeding their initial wealth. However, that is actually what we saw during the IT bubble, with leveraged investors such as hedge funds speculating on further price increases.



6



Dynamic Real-time Strategy



The results in the previous section use the complete sample of observations. To confirm that our conclusions are also valid for “real” investors, we simulate a simple dynamic realtime strategy. This strategy uses only information that is available to investors at the 18



specific point in time. At every month t, the investor uses the bubble detection method outlined in section 3 to test whether an industry experiences a bubble. If she detects a bubble, she decides to invest in that industry for the following month t + 1. If several industries experience a bubble, then we apply the 1/N rule. Although this approach is simplistic, DeMiguel, Garlappi, and Uppal (2009) show that it performs comparably to the sample-based mean-variance model and its extensions. If no industry experiences a bubble, we do not make an investment. Table 4 shows the descriptive statistics for this strategy. Because bubbles are rare events, we find that even with 48 assets, in about 30% of our sample the investor does not observe a bubble in any industry. If she learns about a bubble in certain months, then the average number of bubbles is 3.5 for the CAPM and about 2.5 for the 3F- and 4F-Models. During the months in which the investor makes an investment, she earns an annualized return of 8% based on the CAPM. For the 3F- and 4F-Models, the abnormal return is lower but still an economically sizable 3%-4% per year. The overall positive return on this strategy can be explained by the fact that the strategy has more gaining months than loosing months. In addition, the average gain is in absolute terms slightly larger than the average loss during a loosing month. [Table 4 about here.] In Figure 3, we display the cumulative abnormal log returns to the riding bubble strategy beginning in July 1936 for the CAPM and the 3F-Model and in January 1937 for the 4F-Model. These graphs also reveal that riding bubbles is a risky strategy. An investor who rides bubbles must also be prepared to face, at least temporarily, large losses as the bubbles deflate. However, on average, she can earn a substantial return. [Figure 3 about here.]



7



Alternative Explanations



Our results are consistent with the riding bubbles hypothesis put forward by De Long, Shleifer, Summers, and Waldmann (1990b) and Abreu and Brunnermeier (2003). In this section, we investigate whether alternative theories can explain our findings. 19



7.1



Industry Momentum



We investigate whether industry momentum can serve as an explanation for the bubbles and the subsequent positive abnormal returns. Because bubbles are by definition positive, we focus on a comparison between bubbles and buy-side momentum (i.e., the long positions or “winner” portfolios). Throughout our study, we account for return momentum by including the Carhart (1997) momentum factor in Equations (1) and (4). The Carhart (1997) factor might not be able to completely explain return momentum in the industry portfolios because stock momentum does not entirely subsume industry momentum (Moskowitz and Grinblatt (1999)). Therefore, it is important to analyze carefully the similarities and differences of both strategies. One obvious difference between the two strategies is their time horizon. Table 1 shows that the average length of a bubble is about 30 months. Over this time interval, buy-side industry momentum, as reported by Moskowitz and Grinblatt (1999), actually reverses and profits become negative. This finding accords with Fama and French (1988), who report a negative autocorrelation for industry returns at similar horizons. This negative correlation contrasts our finding of positive abnormal returns following bubbles. We examine the relation between industry bubbles and industry momentum more formally along two lines. In our first analysis, we compare the industries selected by a momentum strategy and an industry bubble strategy. Although there is some overlap, we find that the strategies are substantially different. Second, we investigate whether an industry momentum factor can explain our findings. Even if the two strategies contain different assets, they could both correlate with the same latent risk factor. In this case, the industry momentum factor might be able to explain our findings. We construct industry momentum portfolios based on our sample of the 48 industry portfolios given by Fama and French (1997). Moskowitz and Grinblatt (1999) use 20 industry portfolios in their study. They allocate the top (bottom) three industries to the winner (loser) portfolio, which corresponds to 15% of their sample. For our sample, 15% translates into including the top (bottom) seven (48 × 0.15 ≈ 7) industries in the winner (loser) portfolio. Following Moskowitz and Grinblatt (1999), we construct a six-month, six-month momentum factor with a one-month lag. Since we require a minimum length 20



of 12 months for the bubbles, we also build a momentum portfolio based on a 12-month ranking period, a one-month lag, and an investment during the following month. We compare the industry bubble strategy and the industry momentum strategy by means of a classification table, also known as prediction/realization table. We are interested in whether the industries experiencing bubbles are also included in the momentum portfolio. Therefore, we interpret the industries selected by a buy-side momentum strategy as predictions, and the industries selected by the bubble strategy as realizations. We report classification tables for the different models and the six-month and twelvemonth momentum strategies in Table 5. In case of the CAPM, the total number of industries invested in over time by a bubble strategy is 2071. In 934 cases, these industries are also selected by a momentum strategy and in 792 cases they are selected by a 6month momentum strategy. This implies that success rate, formally known as sensitivity, of the 12-month momentum strategy is 934/2071 = 45%. The sensitivity of the six-month strategy is 38% and thus a little lower. For the abnormal returns based on the 3-Factor Model and 4-Factor Model, the sensitivities vary from 43% to 51%. To be able interpret these figures, we have computed the upper and lower bounds. If the momentum strategy had no predictive power whatsoever, it would predict, by construction, 15% of the industry bubbles correctly. The explanation is that the winner momentum portfolio comprises seven industries which is about 15% of the sample. If the winner momentum strategy was a perfect bubble predictor, the resulting portfolio should, in theory, include all industry bubbles. In our sample this is not possible since we sometimes identify more than seven bubbles. Consequently, the sensitivity of the maximum possible subset is, in particular for the CAPM, slightly below one. Comparing the success rate of the momentum strategy to these upper and lower limits shows that it is not a good predictor of industry bubbles. The momentum strategy has some predictive power because it performs better than a random selection strategy. However, an investor using a winner momentum strategy as a simplified bubble detection method would miss about 50% or more of the industry bubbles. We therefore conclude that our bubble detection method contains incremental information beyond buy-side industry momentum and cannot be replicated by simply following such a strategy.



21



[Table 5 about here.] We also investigate the specificity, which is the ratio of the number of industries that are not included in the buy-side momentum portfolio over the number of industries that do not experience a bubble. Thus, this statistic reflects the extent to which the buy-side momentum portfolio includes industries that have no bubble, i.e. the momentum strategy’s missed shots. In case buy-side momentum was unrelated to bubbles, the expected value would be 85%. Again, this is by construction because 85% of all industries are not included in the momentum portfolio. In the maximum subset case, all bubbly industries (with a maximum of 7) are included in the momentum portfolio, but still, not all industries in the momentum portfolio necessarily experience a bubble. Because the momentum strategy mostly contains more assets than the bubble strategy, the maximum specificity is around 0.88 and 0.89. Across the different models and industry momentum definitions, the specificity of our analysis is about 86%. It is much closer to a scenario in which momentum has no predictive power for industry bubbles than the upper limit. To get a better impression of the economic implications and how the portfolios actually look like, we display the sensitivity and specificity on a monthly basis in the bottom panel of Table 5. Although the results vary somewhat per model and momentum definition, the general impression is that the momentum portfolio contains approximately one industry bubble (or often less) and about six (or slightly more) industries that do not experience bubble. Since there are around two bubbles per month, it means that the momentum strategy misses about one bubble per month. We also examine whether an industry momentum factor can explain our findings. Even though the bubble and buy-side momentum strategy do not contain the same assets, they might both correlate with the same latent risk factor. In this case, even if the portfolios contain different assets, the industry momentum factor should be able to explain the abnormal returns of the bubble strategy. To examine whether an industry momentum factor can explain the abnormal returns following bubbles, we replace the Carhart (1997) momentum factor in our basic models in Equation (1) and Equation (4) by one of the two industry momentum factors constructed above. Table 6 shows that our findings for the risk-return tradeoff do not change substantially. They are similar to the results for the conventional 22



4F-Model presented throughout the paper. The mean and median abnormal returns after the detection of a bubble are significantly larger than if no bubble is detected. Risk, which we measure by the volatility and downside risk, also increases when a bubble appears. Because our findings for the risk-return trade-off using the industry momentum factors are similar to our previous findings, we expect that our results for the asset allocation decision will be similar as well. Table 7 confirms this reasoning. The weight allocated to the risky asset increases significantly upon bubble detection. This increase translates into a significant increase in utility when a bubble is detected, and a positive certainty equivalent. Overall, we conclude that industry momentum is a phenomenon that is different from the the industry bubbles, and that an industry momentum factor does not subsume abnormal returns following bubbles. [Table 6 about here.] [Table 7 about here.]



7.2



Good News Reported in the Media



Another potential explanation for our findings is good news reported in the media. Chan (2003) analyzes the stock market reaction to news releases and finds that it is limited to a couple of months. Thus, the reaction to a single positive news item is much shorter than the minimum length of the industry bubbles. However, perhaps there is a stream of good news, which, especially if the market underreacts to the new information, may cause a prolonged period of positive abnormal returns. We call this phenomenon a “rally”. Although it is not possible to distinguish between bubbles and rallies ex ante, we can expect a different pattern ex post. During a rally, the asset is initially undervalued and the price rises to become equal to fundamental value. Once the asset is no longer substantially undervalued, abnormal returns should, on average, be close to zero. For bubbles, the opposite should hold. The asset is overvalued and the price will ultimately return to fundamental value. To examine whether rallies are a feasible explanation for our findings, we investigate the returns following bubbles. Again, we define one bubble as a sequence of uninterrupted 23



detections. Figure 4 presents the monthly abnormal returns for two years after a bubble. For all three asset pricing models, we observe a large negative return in the first month after the bubble. Since bubbles end with crashes, this finding is not surprising. After the initial very negative return we see a continuous price deflation over the next one to two years. Twelve months after the end of a bubble, the cumulative abnormal returns are -11.9% for the CAPM, -10.14% for the 3F-Model, and -9.06% for the 4F-Model. During the second year, the decline slows down. Two years after a bubble, cumulative abnormal returns are -14.77% for the CAPM, -13.45% for the 3F-Model, and -10.82% for the 4FModel. Overall, we conclude that on average, returns following bubbles are very negative. We cannot rule out that not a single one of the bubbles we identify is truly a rally, but this finding indicates that the bubbles are generally distinct from rallies. [Figure 4 about here.]



7.3



Structural Breaks in Risk Factors



Another explanation for our results could be that the asset pricing model is misspecified, as we do not account for structural breaks in risk factors. An omitted structural break in any of the risk-factor exposures could translate into an estimated structural break in ατ . We might wrongly classify these structural breaks as bubbles. In such a scenario, we would attribute the positive abnormal returns following bubbles to the higher systematic risk not captured by the asset pricing model. We investigate whether such scenario is a plausible alternative explanation for our findings. We formally derive the effect of latent structural breaks in the risk-factor coefficients on our bubble identification, estimate the relevant parameters from our sample, and conclude that to lead to a misclassification as bubble, a structural break in any of the risk-factor exposures must be implausibly large. During our sample period, we do not observe any changes in exposures of the size needed for misclassification. Formally, to derive the effect of a structural break in any of the risk-factor coefficients on our bubble identification method, we assume that the following expected return model is the true model: ˜ rτ = α + βτ f1τ + γ ′ fτ + ετ , E[ετ ], E[ε2 ] = σ 2 τ 24 τ = t − T + 1, . . . , t. (10)



We assume that a structural break in the coefficient for the risk factor f1τ is present   β1 for τ = t − T + 1, . . . , t − ζ , (11) βτ =  β2 for τ = t − ζ + 1, . . . , t ˜ where 0 β1 , and E[f1 ] > 0, the asymptotic bias will be positive. An omitted structural break not only affects the asymptotic value of the difference in intercepts, but also the variance: T Var[ˆ 2 − α1 ] → (β2 − β1 )2 Var[f1 ] + α ˆ 1 σ2, ξ(1 − ξ) (13)



where ξ = 1 − ζ/T is the fraction of observations before the structural break. An omitted structural break in the risk-factor coefficients biases the variance upwards. The test statistic for the structural break in α is the ratio of the difference between α1 and α2 to the variance of this difference. Using Equations (12) and (13), its bias will be √ T (β2 − β1 ) E[f1 ] . (14) χSBF = √ 1 (β2 − β1 )2 Var[f1 ] + ξ(1−ξ) σ 2 Equation (14) shows that both the numerator and the denominator of this t-statistic increase because of the misspecification. Since the increases in the numerator and the denominator depend on the expected value E[f1 ] and the variance Var[f1 ] of the risk factor, the combined effect is an empirical question. 25



Therefore, we start by computing the long-run average and variance of each risk factor. To determine the range of interest, we determine the largest increases in the factor exposures over a period of ten years for all industries based on the 4F-Model. We find maximal increases of 1.16 in the exposure to the market return, 2.7 for SMB, 2.16 for HML, and 1.31 for MOM. Then we compute the resulting test statistic for structural breaks of a varying size. We assume that the structural break is located exactly in the middle of the test period (i.e., ξ = 1/2) because the bias is maximized for this setting (see Appendix B.1). We set the residual variance equal to the overall average of 4%. Figure 5(a) shows the bias in the χ statistic as a function of the true structural break size in β. The solid line corresponds to the case in which the structural break is in the exposure to the market factor. We observe that the bias in χ increases if the true structural break becomes larger. For a break in the CAPM-β of 0.5, the bias equals 0.42. If the break becomes larger, for example one, then the bias rises to 0.74. For a confidence level of 97.5%, the critical value given by Andrews (1993) is 2.82. Even for improbably large structural breaks exceeding two, the bias in χ is well below this value. We also display the bias for the other factors. The bias due to breaks in the exposure to the size factor SMB or value factor HML are well below those for the market return. A break in the exposure to the momentum factor can have a larger the effect, but this bias is also well below the critical value of 2.82. [Figure 5 about here.] The long-run average may understate the potential bias. We repeat the analysis using the largest ten-year average of each risk factor in our sample. Figure 5(b) shows the relation between bias and the true structural break size for this subset. A break in the exposure to the market return produces the largest bias. Over the period July 1949 to June 1959, the average market return equals 1.47% per month, with a volatility of only 3.2%. If the structural break in CAPM-β is 0.5, then the bias would be 0.98. For a break of one, the bias would be 1.87. Only for implausibly large breaks exceeding 1.75 would the bias exceed the critical value of 2.82. Based on this analysis, we conclude that it is very unlikely that our bubble detection method would identify a bubble when an industry is actually exhibiting a structural change 26



in its exposure to a risk factor. Even when we choose the parameter values such that the effect of misspecification is maximum, we need an improbably large structural break in the factor exposure to surpass the critical values. We conclude that structural breaks in factor exposures do not qualify as an alternative explanation for our findings.



7.4



Omitted Risk Factors



Here, we investigate whether an omitted risk factor could explain our results. In such a scenario, the bubbles we identify might be an unknown, omitted risk factor rather than a mispricing. Consequently, the positive abnormal returns following the “bubbles” would truly be a compensation for the omitted risk factor. We derive the effect of an omitted risk factor on our bubble definition in Appendix B.2. In this section we discuss the main results. The true underlying model is similar to Equation (10): rτ = α + βτ f1τ + γf2τ + ετ , E[ετ ], E[ε2 ] = σ 2 τ τ = t − T + 1, . . . , t. (15)



We assume without loss of generality that the model contains two factors, and that the first factor is omitted in the estimation. The exposure to this factor might be constant, i.e., βτ = β or show a structural break as in Equation (11). Instead of the true model in Equation (10), we omit the first factor and estimate the following model rτ = ατ + γ f2τ + et , E[˜t ] = 0 ˜ ˜ ˜ e   α1 for τ = t − T + 1, . . . , t − ζ ˜ αt = ˜  α2 for τ = t − ζ + 1, . . . , t. ˜



(16)



Again, we look at the bias in the structural break in ατ . In Appendix B.2 we derive ˜ the asymptotic difference between a1 and a2 ˜ ˜ plimT →∞ α2 − α1 = (β2 − β1 ) E[f1 ]. ˜ ˜ (17)



This expression is equal to our result in the previous section, which shows that the limiting bias in the difference is the product of the size of the structural break and the average of the omitted factor. This expression shows that an omitted factor influences our detection 27



method only if the asset exhibits a structural break in its exposure towards this factor. If an omitted factor does not show a structural break, then the effect on the constant is the same for a1 and a2 , and the effects cancel out when we take a difference. Therefore, a ˜ ˜ constant exposure to an omitted risk factor cannot be mistaken as a bubble. We conclude that omitted factors without a structural break cannot be an explanation for our findings. If the omitted risk factor exhibits a structural break, then its effect on our bubble identification is much like the effect of an included risk factor for which we omit only the break. The only difference is that since we need to include the additional error of missing out on the factor completely, the increase in the residual variance is larger. A larger residual variance leads to a larger variance of the estimated difference between a1 and a2 . In the ˜ ˜ Appendix B.2 we prove that χOBF = α2 − α1 ˜ ˜ (β2 − β1 ) E[f1 ] (β2 − β1 ) E[f1 ] = ≤ = χSBF . Var[˜ 2 − α1 ] α ˜ Var[˜ 2 − α1 ] α ˜ Var[ˆ 2 − α1 ] α ˆ (18)



Equation (18) shows that the effect of a structural break in a factor exposure on the test statistic is actually smaller when the factor is omitted than when it is included in the asset pricing model. Hence, as long as omitted factors have characteristics (i.e., means and variances) similar to the typical factors used in asset pricing, they cannot explain our results.



8



Robustness Checks



In designing our analysis, we make a number of arbitrary choices. To analyze whether these choices affect our conclusions, we replicate our main findings for several different settings and analyze whether our results are robust to a variety of specifications.



8.1



Significance Level of Bubble Definition



We modify the significance level of the structural break test in Equation (2) and Equation (3), and the subsequent t-test on whether α2 . We raise the confidence level to 99% and then lower it to 95%. We expect to find stronger bubbles, which would be indicated by higher positive abnormal returns if we tighten the definition. The effect on the returns 28



after bubbles and the allocation is ambiguous. Theoretically, perhaps stronger bubbles are more likely to reach the maximum size and burst sooner. This reasoning implies that we should expect future returns to be lower and more volatile after stronger bubbles. Consequently, the optimal weight should be lower after stronger bubbles. However, it is also possible that stronger bubbles are associated with higher future abnormal returns while they continue developing. Table 8 shows the summary statistics of the abnormal returns with and without a bubble for different confidence levels. For all different settings, the standardized abnormal returns are significantly larger when a bubble is detected than if no bubble is detected. Sometimes, the returns are slightly higher and at other times slightly lower, but overall they are similar to our standard specification in Table 2. There is also no clear pattern based on different confidence levels. For the volatility estimates, there are only minor deviations from our original results. As in our previous results, we find that volatility is economically and statistically larger when a bubble is detected than if no bubble is detected. It seems that if we make the tests stricter, the volatility after bubbles increases slightly, but the differences are small. The downside risk measures Value-at-Risk (VaR) and Expected Shortfall (ES) do not show a clear pattern. In most cases, downside risk is more pronounced after bubble detection but the difference is not always statistically significant. Only for the 4F-Model are the differences in downside risk with and without prior bubble detection consistently significant. [Table 8 about here.] Since our findings for the distribution of abnormal returns with and without bubble detection are similar for the different confidence levels, we expect to observe the same for the asset allocation. Table 9 presents the investor’s asset allocation for different confidence levels. For all different confidence levels and asset pricing models, the increase in the optimal weight is economically and statistically significant. This finding confirms our previous conclusion that riding bubbles is optimal. In line with the optimal weights and our previous results, the expected utility is also consistently larger after bubble detection and the certainty equivalent is always significantly positive. [Table 9 about here.] 29



8.2



Bubble Length and Estimation Period



Two other arbitrary choices in our bubble definition are the length of the bubble and the estimation period. To examine the robustness of our results to these choices, we start by modifying the maximum length of the bubble per detection. Table 10 compares the risk and return trade-off if no bubble is detected and following bubble detection for bubbles that may last either at most for three years or up to seven years. Formally, we allow ζ in Equation (3) to vary from 12 to 36 and 12 to 84 while keeping the estimation period T at 120 months. In the standard setting, ζ could vary from 12 to 60. We also investigate the robustness of our findings to the choice of estimation period and halve the estimation period (i.e., we set T = 60) for bubbles that may last up to three years. The mean abnormal returns after bubble detection are in all cases significantly higher than if no bubble is detected. The magnitude of the returns is similar across the different estimation settings. For example, for the 4F-Model, the mean abnormal return is 0.09 after bubble detection for bubbles that may last three, five, or seven years. The mean abnormal return increases slightly to 0.12 if we restrict the estimation period to five years, i.e., T = 60. As with our previous results, the volatility estimates are consistently higher after bubble detection than if no bubble is detected. The magnitude of the volatility is very similar across the different specifications. We conclude that the abnormal returns after bubbles are not sensitive to the choice of bubble horizon or estimation period. [Table 10 about here.] The same holds for the investor’s asset allocation presented in Table 11. The increase in optimal weight, when a bubble is detected, is in most specifications significant at the 5% level. Only for the 4F-Model specification in which we use a ten-year estimation window and three-year bubble length, is the p-value of the difference in weight 6.6%, and thus slightly larger than the usual 5% significance level. However, even in this specification, the p-values for the difference in utility and the certainty equivalent are clearly below the 5% level. As with our previous findings, the increase in optimal weight is somewhat extreme for the CAPM. For the 3F- and 4F-Models the changes in weight are more realistic. For example, for bubbles that last up to seven years, the optimal weight increases from zero 30



to 0.96 for abnormal returns based on the 3F-Model, and rises from 0.1 to 0.78 for the 4F-Model returns. The changes in utility and the certainty equivalent return confirm our findings for the weights and previous results. [Table 11 about here.]



8.3



Changes in Crash Definition



In our bubble definition, we assume that a bubble ends if, during the previous six months, the investor has detected a crash that is at least twice the standard deviation of abnormal returns. Both the size and horizon of the crash size seem like reasonable choices, but ultimately, they are arbitrary. Therefore, in this section we show that our results are robust to modifying these parameters. In Table 12 we replicate the bubble’s risk and return trade-offs for two different crash windows. In the first case, we set the window to zero, implying that we do not take crashes into account. In the second case, we double the length of the crash window and set it equal to 12 months, which is also the minimum bubble length. The abnormal return appears to decline slightly as the crash window shortens. At the same time, it seems that the variance following bubbles is negatively related to the length of the crash window. This potentially negative relation between the crash window length and the risk-return trade-off could be explained by the fact that crashes tend to cluster.5 Most importantly, we confirm our previous findings. For all specifications, the mean abnormal return following a bubble is significantly higher than if we find no bubble. Even if we do not take crashes into account at all, the monthly standardized abnormal return increases from 0.01 in the absence of a bubble to 0.19 after bubble detection for the CAPM. For the 3F-Model, the return rises from zero to 0.09, and for the 4F-Model it increases from 0.01 to 0.08. The volatility estimates are also consistently larger following bubbles than if no bubble is detected. The difference is for all specifications significant at conventional levels. The downside risk measures, Value-at-Risk and Expected Shortfall are larger following a bubble for most settings, but the difference is not always significant.

5



In an earlier version of this paper, we find that the probability of a crash is higher if there was a crash



during the previous months.



31



[Table 12 about here.] Table 13 shows that when the investor detects a bubble, she substantially increases the weight assigned to the risky asset. The increase in weight is economically and statistically significant for all settings. In line with our findings for the risk-return trade-off, it seems that when a bubble is detected, the optimal weight allocated to the risky asset is positively related to the length of the crash window. However, even if we set the length to zero and thereby effectively do not consider crashes at all, the increase in the optimal weight allocated to the risky asset upon bubble detection is substantial. For the abnormal returns based on the CAPM, the optimal weight increases from 0.06 in the absence of a bubble to 1.6 following the detection of a bubble. The changes in optimal weight are more realistic but still sizable for the 3F- and 4F-Models. For the 3F-Model, the optimal weight rises from zero to 0.78 and for the 4F-Model, it changes from 0.1 to 0.62. Overall, we conclude that riding bubbles is the optimal strategy regardless of whether the investor considers crashes or at which horizon. Our results for the expected utility and the certainty equivalent support this conclusion. [Table 13 about here.] Table 14 presents the risk-return trade-off with and without a bubble for different crash sizes. We apply a stricter boundary of 2.25 times the standard deviation of abnormal returns. Based on a normal distribution, this boundary corresponds to about 1% of return observations. We also loosen the restriction and include crashes up 1.75 times the standard deviation of abnormal returns. This restriction corresponds to about 4% of observations based on the normal distribution. The results for the different crash definitions confirm our original findings. The returns following a bubble are consistently larger than if no bubble is detected. The magnitude of the difference is similar for different crash sizes. As with our previous results, the volatility estimates are always larger when a bubble is detected than if no bubble is detected. For the 3F- and the 4F-Models, downside risk as measured by VaR and ES is more prevalent following bubbles than if no bubble has been detected. [Table 14 about here.] 32



Table 15 confirms that it is optimal for an investor to ride bubbles, regardless of crash size. Across all specifications, we consistently find that the optimal weight increases significantly after the identification of a bubble. The magnitude of the change in weight is similar across the different specifications of crash size. The investor’s utility consistently rises if she detects a bubble. The certainty equivalent the investor demands is consistently positive and statistically significantly different from zero for all crash boundaries. [Table 15 about here.]



8.4



Changes in the Risk-free Rate



In the analysis of the optimal portfolio, we assume that the risk-free rate equals its long-run average. In this section, we investigate how this assumption affects our results. Again, we use the approximation for the optimality condition: [ ] [ ] ∗ ∗ 0 = E[(1 + rf + w∗ ηi )−γ ηi ] ≈ E e−γ(rf +w ηi ) ηi = e−γrf E e−γw ηi ηi . If the approximation is exact, then the risk-free rate is irrelevant, because the optimality [ ] ∗ condition is completely determined by the second part, E e−γw ηi ηi = 0. If the approximation is almost exact, as in our case with monthly abnormal returns, then the choice of the risk-free rate will have only a small influence. As alternative specifications, we have used a risk-free rate of zero and a rate that is twice the long-run average. For both settings, we use our base case risk-aversion level of two. The impact of the changes of the risk-free rate on our results is limited to the third decimal of the estimates. The results are virtually identical, so they are available on request.



8.5



Alternative Utility Functions



To examine the robustness of our results for the power-utility function used in section 5, we replicate our results for investors who have different utility functions.



33



8.5.1



Skewness, Kurtosis, and Mean-Variance Preferences



We investigate whether and how our findings change if an investor’s preferences in skewness and kurtosis differ from the preferences of a power-utility investor. To obtain these results, we begin with a more general utility function based on the different moments of the return distribution, then substitute the parameters implied by the power-utility function for the variance, skewness, and kurtosis. To examine the sensitivity of our findings to these moments, we vary the skewness and kurtosis parameters. Finally, we set the parameters equal to the ones of a mean-variance utility function. We approximate the utility function of an investor a Taylor expansion around his ref¯ erence point of wealth Wt+1 (see for instance Harvey and Siddique, 2000; Jondeau and Rockinger, 2006; Guidolin and Timmermann, 2008):

∞ ∑ 1 ∂ k U (Wt+1 ) ¯ ¯ (Wt+1 − Wt+1 )k . U (Wt+1 ) = k k! ∂Wt+1 k=0



(19)



¯ where we assume that Wt+1 = Wt as in Harvey and Siddique (2000) and consequently Wt+1 = Wt (1 + rf + wηi ). We apply a Taylor expansion up to the fourth order: 1 U (Wt+1 ) =U (Wt ) + U (1) (Wt )Wt (rf + wηi ) + U (2) (Wt )Wt2 (rf + wηi )2 + 2 1 (3) 1 (4) 4 U (Wt )Wt3 (rf + wηi )3 + U (Wt )Wt4 (rf + wηi )4 + O(Wt+1 ), 6 24



(20)



4 where O(Wt+1 ) contains the higher-order terms. When we take expectations, the resulting



equation shows how higher-order moments of the portfolio return enter the utility function: E[U (W )] ≈ κ0 +κ1 E[rf +wηi ]+κ2 E[(rf +wηi )2 ]+κ3 E[(rf +wηi )3 ]+κ4 E[(rf +wηi )4 ], (21) with κk = U (k) (Wt )Wtk /k!. The optimal weight allocated to the risky asset wH to maximize this expression should satisfy: E[ηi ] + 2 ] ] ] κ3 [ κ4 [ κ2 [ E (rf + wH ηi )ηi + 3 E (rf + wH ηi )2 ηi + 4 E (rf + wH ηi )3 ηi = 0. (22) κ1 κ1 κ1



This equation implies that κ0 does not affect wH , so we ignore it. Dividing by κ1 allows us to eliminate this coefficient as well. If the utility function is not known, then the values of the parameters κk are not determined. Necessary and desirable properties of utility functions, such as risk aversion 34



and decreasing absolute risk aversion, restrict only the signs of the parameters: the uneven parameters are positive and the even parameters are negative (see Scott and Horvath, 1980). Therefore, we define κ′2 = −κ′2 /κ1 , κ′3 = κ′3 /κ1 and κ′4 = −κ′4 /κ1 , which yields [ ] [ ] [ ] E [U ′ (W )] = E[rf +wηi ]−κ′2 E (rf + wηi )2 +κ′3 E (rf + wηi )3 −κ′4 E (rf + wηi )4 , (23) where all coefficients should be positive. We then substitute the parameters implied by the power-utility function with γ = 2 in section 5: κ1 = W 1−γ , κ′2 = γ/2, κ′3 = (1 + γ)γ/6, κ′4 = (2 + γ)(1 + γ)γ/24 and solve for the optimal weight using numerical techniques. The first rows of Table 16, in Panels A, B, and C show the optimal weights, the expected utilities, and the certainty equivalent for the three different asset pricing models. We find that the results are consistently similar to the comparable scenario for the exact specification of the power-utility function in Table 3, indicating that a fourth-order approximation is reasonably precise. We vary the investor’s preferences for skewness and kurtosis as we move down each panel. As κ′3 increases, the investor becomes more averse to negative skewness and has a stronger preference for positive skewness. In that case, when she detects a bubble, her allocation to the risky asset increases, because the abnormal returns after bubbles are somewhat positively skewed. For the CAPM the increases are quite pronounced, but for the 3F- and 4F-Models they take on more moderate values and the effects seem more realistic. Accordingly, there is also an increase in both, the investor’s utility and the certainty equivalent she demands. If the investor becomes more concerned about kurtosis and κ′4 rises, then the weight allocated to the bubbly asset decreases slightly. For example, if kurtosis doubles from one to two, then the optimal weight for the CAPM decreases from 1.68 to 1.57. For the 3F- and 4F-Models, the optimal weight decreases from 0.91 to 0.89 and from 0.74 to 0.73, respectively. In all cases, the decrease is not only small, but more importantly, when the investor spots a bubble, the weight allocated to the risky asset is economically and statistically significantly larger than if she does not detect a bubble. The significant increases in utility when a bubble is detected and the positive and significant certainty equivalent confirm our conclusions for the weights. [Table 16 about here.] 35



Finally, we also investigate the optimal allocation of a mean-variance investor by setting κ′3 and κ′4 equal to zero. We set κ′2 equal to one and two, which implies risk-aversion levels of two and four. For the power-utility investor, the optimal weigh declines as risk-aversion increases. However, even with a risk aversion of four, the optimal weight allocated to the risky asset is much larger if a bubble is detected than if no bubble is detected. For the CAPM, the optimal weight is 0.8 if a bubble is detected compared to 0.05 in the absence of a bubble. For the 3F- and 4F-Models, if a bubble is detected, then the optimal weight increases from -0.01 to 0.44 and 0.05 to 0.36. These increases are not only economically large but also statistically significant. The same holds for the increases in utility and the positive certainty equivalent. Overall, these results confirm our earlier findings for the power-utility investors. In addition to power-utility investors, investors who care more about skewness and kurtosis and mean-variance investors would ride bubbles. 8.5.2 Mean - Semivariance Utility



The abnormal returns after bubbles exhibit higher downside risk, which is indicated by the VaR and ES measures in Table 2 in section 4. Figure 2 confirms this finding by showing a fatter left tail following a bubble than if there had been no bubble. Given this evidence of a higher downside risk after bubbles, we investigate whether our conclusions are generalizable to investors who are particularly sensitive to losses. We choose an investor with a mean-semivariance utility function U SV . In contrast to utility functions featuring VaR or ES, it has the desirable property that it is concave for losses. Following Harlow and Rao (1989), we link the utility function to the portfolio return rp :   rp − γ(k − rp )2  rp for rp ≤ k for rp > k,



U SV (rp ; k) =



(24)



where k is the target return. Realizations of a portfolio return below k lead to a discount of utility beyond the realized return. We follow Bawa and Lindenberg (1977) and assume that the target return equals the risk-free rate. 36



The second term of Equation (24) leads to the semi-variance with respect to k: ∫ k SVk [rp ] = (k − rp )2 dF (rp ),

−∞



(25)



where F (rp ) is the cumulative distribution function of rp . The investor combines this utility function with the expression for the portfolio return in Equation 5 and solves: [ ] max E U SV (rf + wηi ; rf ) = max{rf + w E[ηi ] − γSV0 [wηi ]}.

w w



(26)



Assuming that the threshold k equals the risk-free rate implies considering the semivariance of the risky part of the portfolio with respect to zero, SV0 [wηi ]. Due to this result, zero becomes the investor’s variance threshold, and we can write SV0 [wηi ] = w2 SV0 [sgn(w) · ηi ]. Solving for the optimal weight wSV leads to wSV = E[ηi ] , 2γSV0 [sgn(E[ηi ]) · ηi ] (27)



where we replace SV0 [sgn(w) · ηi ] by SV0 [sgn(E[ηi ]) · ηi ], because the sign of the weight is solely determined by the sign of the expected abnormal return. Table 17 shows the optimal weight, the expected utility, and the certainty equivalent for an investor with a risk-aversion coefficient equal to two. The optimal weight allocated to the risky asset increases substantially when the investor detects a bubble. For the CAPM, it rises from 0.1 in the absence of a bubble to an amazing 2.39 in the presence of a bubble. Thus, the investor would take a position of more than twice his total position. The allocations for the 3F- and the 4F-Models are more reasonable. For abnormal returns based on the 3F-Model, the weight in the risky asset increases from -0.02 to 1.08 when we detect a bubble, and if we consider the 4F-Model, the weight rises from 0.11 to 0.85. In all cases, when a bubble is detected the increase in the optimal weight is statistically significant. Consistent with the results for the optimal weight, we find that expected utility is higher following a bubble. The certainty equivalent demanded by the investor for not updating his portfolio is always positive, sizable, and statistically significantly different from zero. Overall, we conclude that even for an investor who is particularly concerned with downside risk, riding bubbles is the optimal strategy. [Table 17 about here.] 37



9



Summary and Conclusion



The optimal response of a rational investor to asset price bubbles is a crucial factor in explaining the existence and persistence of bubbles. In this study, we use an empirical and normative approach: Instead of making specific assumptions on the characteristics of bubbles, we base our evidence on bubbles observed in U.S. industry portfolios. We require that the investor infers the presence of a bubble from publicly available information. Although this approach results in a noisy bubble identification, it describes a real-world setting and is applicable by many investors. In contrast to other empirical studies, our study is not limited to a specific type of investor or to a specific historical bubble episode. To find out how a bubble affects an investor’s optimal portfolio, we develop a new bubble detection method. We identify a bubble if a structural break in the α of an asset pricing model is present in the recent past, after which α becomes significantly positive. The bubble ends when a crash takes place. This approach enables us to split the set of returns in those for which we have, and those for which we have not, found bubbles beforehand. We consider the CAPM, the Fama and French (1993) three-factor model and Carhart (1997)’s four-factor model as asset pricing models. The abnormal returns following the detection of a bubble offer a favorable risk-return trade-off. A rational investor equipped with a power-utility function will increase her investment significantly, when she learns about a bubble. We conclude that the additional average abnormal return after bubble detection more than compensates for the rise in risk. A simple dynamic bubble-riding strategy that uses only real-time information earns positive abnormal returns in the order of 3% to 8%. We evaluate several alternative explanations for these findings. The industry bubbles are a different phenomenon from the industry momentum described by Moskowitz and Grinblatt (1999). Since the bubbles end with large negative abnormal returns, we infer that they cannot be explained by an underreaction to good news. We derive analytically that the industry bubbles do not result from a misspecification of the asset pricing models used in this study: the bubbles cannot be explained by an omitted risk factor, an omitted structural break, or by a combination thereof.



38



Based on our empirical results, we conclude that riding bubbles is a rational investor’s optimal strategy. If every investor always followed this strategy, bubbles would be explosive and last infinitely. However, we generally do not observe explosive stock price developments. Abreu and Brunnermeier (2003) provide a setting in which this apparent contradiction can be resolved. In their model, arbitrageurs cannot synchronize selling out, so it is optimal for them to ride the bubble as long as it continues. However, synchronizing events can induce a sufficiently large number of rational investors to sell and cause the bubble to burst. These events do not necessarily need to contain any new information on market fundamentals, but could, for example, be large price movements or breaks through psychological resistance lines. The findings of this study contribute to the literature on bubbles by explaining their persistence, reoccurrence and importance. Since it is optimal for rational investors to ride bubbles, these investors are not likely to cause a bubble to burst shortly after it starts. Instead, rational investors have strong incentives to fuel bubbles and thus increase their size. From a practical perspective, our study helps to explain the fact that we have observed sizable bubbles since the inception of financial markets hundreds of years ago until very recently. We can expect to observe bubbles in financial markets in the future as well.



39



References

Abreu, D. and Brunnermeier, M. K. (2003). Bubbles and crashes. Econometrica, 71(1):173–204. Andrews, D. W. K. (1993). Tests for parameter instability and structural change with unknown change point. Econometrica, 61(4):821–856. Bai, J. (1994). Least squares estimation of a shift in linear processes. Journal of Time Series Analysis, 15(5):453–472. Bai, J. and Perron, P. (1998). Estimating and testing linear models with multiple structural changes. Econometrica, 66(1):47–78. Bawa, V. S. and Lindenberg, E. B. (1977). Capital market equilibrium in a mean-lower partial moment framework. Journal of Financial Economics, 5(2):189–200. Blanchard, O. J. and Watson, M. W. (1982). Bubbles, rational expectations and financial markets. In Wachtel, P., editor, Crisis in the Economic and Financial Structure, pages 295–315. Lexington Books, Lexington MA, USA. Brooks, C. and Katsaris, A. (2005). A three-regime model of speculative behaviour: Modelling the evolution of the S&P 500 Composite Index. Economic Journal, 115:767–797. Brunnermeier, M. K. and Nagel, S. (2004). Hedge funds and the technology bubble. Journal of Finance, 59(5):2013–2040. Campbell, J. Y. and Shiller, R. J. (1987). Cointegrating and tests of present value models. Journal of Political Economy, 95(5):1062–1088. Carhart, M. M. (1997). On persistence in mutual fund performance. Journal of Finance, 52(1):57– 82. Chan, W. S. (2003). Stock price reaction to news and no-news: Drift and reversal after headlines. Journal of Financial Economics, 70(2):223–260. Chen, J., Hong, H., and Stein, J. C. (2001). Forecasting crashes: Trading volume, past returns, and conditional skewness in stock prices. Journal of Financial Economics, 61(3):345–381. Cochrane, J. H. (2001). Asset Pricing. Princeton University Press, Princeton, NJ, USA. Dass, N., Massa, M., and Patgiri, R. (2008). Mutual funds and bubbles: The surprising role of contractual incentives. Review of Financial Studies, 21:51–99. De Long, J. B., Shleifer, A., Summers, L. H., and Waldmann, R. J. (1990a). Noise trader risk in financial markets. Journal of Political Economy, 98(4):703–738. De Long, J. B., Shleifer, A., Summers, L. H., and Waldmann, R. J. (1990b). Positive feedback investment strategies and destabilizing rational speculation. Journal of Finance, 45(2):379–395.



40



DeMiguel, V., Garlappi, L., and Uppal, R. (2009). Optimal versus naive diversification: How inefficient is the 1/n portfolio strategy? Review of Financial Studies, 22:1915–1953. Doornik, J. (1998). Object-Oriented Matrix Programming Using Ox Version 2.0. Timberlake Consultants, West Wickham, UK, 2nd edition. Dow, J. and Gorton, G. (1994). Arbitrage chains. Journal of Finance, 49(3):819–849. Estrella, A. and Rodriguez, A. P. (2005). One-sided test for an unknown breakpoint: Theory, computations, and application to monetary policy. Staff Report 232, Federal Reserve Bank of New York, New York, NY, USA. Fama, E. F. (1965). The behavior of stock market prices. Journal of Business, 38(1):34–105. Fama, E. F. and French, K. R. (1988). Permanent and temporary components of stock prices. Journal of Political Economy, 96(2):246–273. Fama, E. F. and French, K. R. (1993). Common risk factors in the returns on stocks and bonds. Journal of Financial Economics, 33(1):3–56. Fama, E. F. and French, K. R. (1997). Industry costs of equity. Journal of Financial Economics, 43(2):153–193. Garber, P. M. (1989). Tulipmania. Journal of Political Economy, 97(3):535–560. Goldman, E. and Slezak, S. L. (2003). Delegated portfolio management and rational prolonged mispricing. Journal of Finance, 58(1):283–311. Greenwood, R. and Nagel, S. (2009). Inexperienced investors and bubbles. Journal of Financial Economics, 93(2):239–258. Griffin, J. M., Harris, J. H., and Topaloglu, S. (2006). Who drove and burst the tech bubble? Working Paper, Queen’s University. Guidolin, M. and Timmermann, A. (2008). International portfolio choice under regime switching, skew and kurtosis preference. Review of Financial Studies, 21(2):889–935. Hansen, B. E. (2001). The new econometrics of structural change: Dating breaks in US labor productivity. Journal of Economic Perspectives, 15(4):117–128. Harlow, W. V. and Rao, R. K. (1989). Asset pricing in a generalized mean-lower partial moment framework: Theory and evidence. Journal of Financial and Quantitative Analysis, 24(3):285– 311. Harvey, C. R. and Siddique, A. (2000). Conditional skewness in asset pricing tests. Journal of Finance, 55(3):1263–1295. Jondeau, E. and Rockinger, M. (2006). Optimal portfolio allocation under higher moments. European Financial Management, 12(1):29–55.



41



Kindleberger, C. P. (2000). Manias, Panics, and Crashes, a History of Financial Crises. John Wiley & Sons, Inc., New York, NJ, USA, 4th edition. Lintner, J. (1965). The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. Review of Economics and Statistics, 47:13–37. Moskowitz, T. J. and Grinblatt, M. (1999). Do industries explain momentum. Journal of Finance, 54(4):1249–1290. Ofek, E. and Richardson, M. (2002). The valuation and market rationality of internet stock prices. Oxford Review of Economic Policy, 18(3):265–287. ˇ P´stor, L. and Veronesi, P. (2006). Was there a NASDAQ bubble in the late 1990s? Journal of a Financial Economics, 81(1):61–100. Scott, R. C. and Horvath, P. A. (1980). On the direction of preference for moments of higher order than the variance. Journal of Finance, 35:915–919. Sharpe, W. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance, 19:425–442. Shiller, R. J. (2000). Irrational Exuberance. Princeton University Press, Princeton NJ, USA. Shleifer, A. and Vishny, R. (1997). The limits of arbitrage. Journal of Finance, 52(1):35–55. Tabachnick, B. G. and Fidell, L. S. (2001). Using Multivariate Statistics. Allyn and Bacon, Boston, MA, USA, fourth edition. Temin, P. and Voth, H.-J. (2004). Riding the South Sea bubble. American Economic Review, 94(5):1654–1668.



42



Table 1: Descriptive Statistics of Bubbles

Mean 30.6 0.468 -0.0061 0.023 3.22 34.5 CAPM Med. 28.3 0.395 -0.0057 0.020 3.16 33.0 SD 23.5 1.16 0.0050 0.012 0.752 15.3 Mean 32.4 0.494 -0.0055 0.021 3.08 31.2 3F-Model Med. SD 31.1 23.0 0.424 1.18 -0.0049 0.0050 0.019 0.012 2.89 0.763 28.0 14.7 Mean 30.6 0.509 -0.0051 0.021 3.07 31.3 4F-Model Med. SD 29.5 22.7 0.439 1.23 -0.0045 0.0049 0.018 0.011 2.94 0.700 28.0 14.9



Raw Return St. Abn. Return α1 α2 Strength Length



This table reports the mean, median (Med.) and standard deviation (SD) of six bubble statistics: raw returns during bubbles (in % per year), standardized abnormal returns during bubbles, α1 , α2 , strength at detection (t-statistic), and length at detection. We regress the industry returns on a constant and the market return (column “CAPM”), Fama and French (1993)’s three factors (column “3F-Model”), or Carhart (1997)’s four factors (column “4F-Model”). If a ten-year series of industry returns shows evidence of an upward structural break in the constant and the constant is significantly positive after the break, an investor detects a bubble. The t-statistic of the constant gives the strength of the bubble. A bubble has ended if a crash has occurred in the last six months, where a crash is defined as a residual below -2 times its standard deviation. Critical values for the structural break test correspond with a 97.5% confidence level, and are obtained from Andrews (1993).



43



Table 2: Standardized Abnormal Returns With and Without Prior Bubble Detection

(a) CAPM No Bubble Detected # Obs. Mean Median Volatility Skewness Kurtosis VaR(0.95) ES(0.95) VaR(0.975) ES(0.975) 36293 0.01 −0.01 1.03 0.18 5.46 1.59 2.23 2.01 2.67 (0.01) (0.00) (0.01) (0.04) (0.20) (0.01) (0.02) (0.02) (0.03) Bubble Detected 2071 0.19 0.14 1.17 0.52 5.80 1.63 2.18 1.96 2.57 (0.03) (0.02) (0.03) (0.21) (1.47) (0.07) (0.07) (0.09) (0.10) p-value ξT, where ξ ∈ (0, 1). So, the true model exhibits a structural break in the exposure to xt instead of the intercept. The first step of our bubble detection method only allows for a structural break in the intercept, so it estimates the model yt = at + bxt + ct wt + et , E[et ] = 0 { a1 for t ≤ ξT at = a2 for t > ξT



(30)



with OLS. To estimate this model, a sample of size T is available, with T1 observations before the structural breakpoint ξT , and T2 observations thereafter. We use Y , X and W to denote the vector of observations and U for the vector of error terms. A subscript 1 (2) denotes the subvectors before (after) the structural break. First we derive the OLS estimates. We define the auxiliary matrix ( ) ıT1 0 X1 W1 ZT = , 0 ıT2 X2 W2



75



where ιm denotes a vector of length m filled with ones. Standard regression theory gives the estimates for the coefficients   a1 ˆ ( ) a2  ( ′ ) ′ ˆ  = ZT ZT −1 ZT Y1 . (31) ˆ Y2 b c ˆ Next, we use asymptotic theory to derive the properties of these estimates. We use mn to x denote the nth moment of the variable xt , and similar for the other variables; and mxw for the comoment of x and w. We assume that the moments of the explanatory variables are constant over time, and do not change with the structural break. We calculate   T1 0 ı′ 1 X1 ı′ 1 W1 T T  0 T2 ı′ 2 X2 ı′ 2 W2  ′ T T  ZT ZT =  ′  ı X1 ı′ X2 X ′ X X ′ W  , T1 T2 ı′ 1 W1 ı′ 2 W2 X ′ W W ′ W T T and use this to define  ξ 0 ξm1 ξm1 x w  0 1 ′ (1 − ξ) (1 − ξ)m1 (1 − ξ)m1  x w ≡ lim ZT ZT =  1 .  ξmx (1 − ξ)m1 m2 mxw  T →∞ T x x ξm1 (1 − ξ)m1 mxw m2 w w w 



Σzz



(32)



In a similar fashion we calculate     ı′ 1 Y1 T1 α + β1 ı′ 1 X1 + γı′ 1 W1 + ı′ 1 U1 T T T T ( )     Y1 ı′ Y T2 α + β2 ı′ 2 X2 + γı′ 2 W2 + ı′ 2 U2 ′ T T T  ZT =  ′ T2 2 ′  =  ′  X Y1 + X Y2   ı Xα + β1 X ′ X1 + β2 X ′ X2 + γX ′ W + X ′ U  , Y2 1 2 1 2 T ′ ′ ′ ′ W1 Y1 + W2 Y2 ı′ W α + β1 X1 W1 + β2 X2 W2 + γW ′ W + W ′ U T where we have substituted the true model for yt . We use this result to define ( )   ξ α ( β1 m1 + γm1 + m1 ) + x w u   1 ′ (1(− ξ) α + β2 m1 + γm1 + m1 x) w u  Σzy ≡ lim ZT YT =   αm1 + ξβ1 + (1 − ξ)β2 m2 + γmxw + mxu  . T →∞ T x ) x ( αm2 + ξβ1 + (1 − ξ)β2 mxw + γm2 + mwu w w Consequently, we find     a1 ˆ α + (1 − ξ)(β1 − β2 )m1 x a2   α − ξ(β1 − β2 )m1  ˆ x . plimT →∞  ˆ  =   b  ξβ1 + (1 − ξ)β2 γ c ˆ



(33)



(34)



As expected, ˆ converges to a weighted average of β1 and β2 , where the weight depends on the b proportions of the sample before and after the structural break. The deviations of a1 and a2 from the true intercept α reflect the size of the structural break β1 − β2 , the proportion ξ and the average value of xt .



76



The structural break test in our bubble detection method also needs the variance of the estimator. Therefore, we derive the variation of the residuals et . We have { (1 − ξ)(β1 − β2 )(xt − m1 ) + υt for t ≤ ξT x et = 1) + υ −ξ(β1 − β2 )(xt − mx for t > ξT t

2 so σe becomes 2 2 2 σe = ξ(1 − ξ)(β1 − β2 )2 σx + συ ,



(35)



2 where σx is the (population) variance of xt . This expression shows that the misspecification leads to an increase in the residual variance. Applying standard regression theory gives the desired result     a1 ˆ α + (1 − ξ)(β1 − β2 )m1 x √ a2   α − ξ(β1 − β2 )m1  ( ) ˆ −1 2 x  → N 0, Σzz σe . (36) T  ˆ  −    b   ξβ1 + (1 − ξ)β2 γ c ˆ



The test statistic for a structural break is based on the difference a2 − a1 , for which we have ˆ ˆ plimT →∞ a2 − a1 = (β2 − β1 )m1 ˆ ˆ x ) ( √ ( ) 1 2 1 σ . T a2 − a1 − (β2 − β1 )mx → N 0, ˆ ˆ ξ(1 − ξ) e This means that the expected value of the statistic for the structural break test on the intercept, when there actually is a structural break in the factor is given by √ √ T ξ(1 − ξ)(β2 − β1 )m1 T (β2 − β1 )m1 x x =√ χSBF = , (37) 2 σe (β − β )2 σ 2 + 1 σ 2

2 1 x ξ(1−ξ) υ



and we conclude that the statistic depends on the average value of the factor, its variance, the residual variance of the returns, the size of the true structural break, and the location of the structural break. We finish this appendix by analyzing the sensitivities of the statistic for the different inputs. As the size and location of the structural break show up in both the numerator and the denominator, we rewrite the statistic as √ ( 2 ( )−1 −2 )−1/2 2 χSBF = m1 T σx + συ ξ − ξ 2 ∆ , x where we use ∆ ≡ β2 − β1 for the size of the structural break. It is straightforward to see that the statistic is increasing in the factor average m1 and in the absolute size of the structural break ∆, x 2 2 and decreasing in the factor and residual variances σx and συ . To find the effect of the location of the structural break, we differentiate χ with respect to ξ: ( )−1 −2 )−3/2 2 ( )−2 −2 dχSBF 1 √ ( 2 2 = m1 T σx + συ ξ − ξ 2 ∆ συ ξ − ξ 2 ∆ (1 − 2ξ). x dξ 2 We conclude that the statistic is maximized for ξ = 1/2, so when the structural break is located in the middle of the sample.



77



B.2



Omitted risk factor



When the risk factor x is omitted, it means that the true model reads as in Equation (15). The factor exposure may be constant, i.e. βt = β or show a structural break as in Equation (29). As the presence of a structural break in β is the more general case (with no break implying β1 = β2 ), we make the derivations under that assumption and discuss subsequently what a constant exposure to an omitted risk factor implies. The first step in our bubble identification procedure allows for a break in the intercept, but would in this case ignore the factor xt . Consequently, it estimates a reduced version of the model in Equation (15): yt = at + ct wt + et , E[et ] = 0 { a1 for t ≤ ξT at = a2 for t > ξT,



(38)



with OLS. The assumptions on the sample and notation are the same as in the previous section. For deriving the estimators, we also follow the same approach as in the previous section. First we define an auxiliary matrix ( ) ıT1 0 W1 ∗ ZT = , 0 ıT2 W2 which is simply ZT without the column X1 , X2 . We use this matrix to construct the coefficient estimates   ( ) a1 ˆ ( ∗ ′ ∗ )−1 ∗ ′ Y1 a2  = ZT ZT ˆ ZT . (39) Y2 c ˆ To derive the asymptotic properties of the estimators, we define two limiting matrices   ξ 0 ξm1 w 1 ∗ ∗ (1 − ξ) (1 − ξ)m1  Σzz ∗ ≡ lim ZT ′ ZT =  0 w T →∞ T 1 1 ξmw (1 − ξ)mw m2 w and Σzy∗ ) (  ξ α ( β1 m1 + γm1 + m1 ) + u w x 1 ∗ . (1 − ξ) α + β2 m1 ) γm1 + m1 ≡ lim ZT ′ YT =  u w x+ ( T →∞ T 2 +m 2 + ξβ + (1 − ξ)β m αmw wu 1 2 xw + γmw  (41)



(40)



We use these two matrices to derive the asymptotic values of the estimators  σxw ¯  α + β1 m1 − 2 βm1   x w σw   a1 ˆ  σxw ¯ 1  a2  = α + β2 m1 − 2 βmw  , plimT →∞ ˆ x   σ   σxw w c ˆ ¯ γ+ 2 β σw



(42)



78



2 where σxw denotes the (population) covariance of xt and wt , σw denotes the (population) variance ¯ = ξβ1 + (1 − ξ)β2 . of wt and we use the shorthand notation β The estimate for c consists of two terms, the true exposure to the factor wt , γ, and a term ˆ that is related to the omitted factor. As could be expected, part of the exposure to the omitted 2 factor comes in via the correlation between xt and wt . The fraction σxw /σw would simply be the ¯ regression coefficient of xt on wt . The factor β = ξβ1 + (1 − ξ)β2 reflects the structural break, and is the weighted average of the factor exposure before and after the structural break. If there is no structural break in the omitted factor, this factor would reduce to β. The estimates for a1 and a2 consist of three terms. The first is the true intercept α. The ˆ ˆ second term shows up because the average of the omitted factor multiplied by its exposure is captured by the intercept. The third term is a correction related to the entrance of the omitted risk factor via wt in c. Therefore, it is simply the second term of c multiplied with the mean of ˆ ˆ wt . Before we derive the variance of the estimators, we first consider the difference between a1 ˆ and a2 : ˆ



plimT →∞ a2 − a1 = (β2 − β1 )m1 . ˆ ˆ x This expression is equal to our result in the previous section, and shows that the limiting bias in the difference is the product of the size of the structural break and the average of the omitted factor. This expression shows that an omitted factor only influences our detection method, if the asset exhibits a structural break in its exposure towards this factor. So, the bubble detection method is not simply driven by constant exposures to omitted factors. As a first step towards the variance of the estimators we consider the residuals  β1 (xt − m1 ) − σxw β(wt − m1 ) + υt for t ≤ ξT ¯  x w 2 σw ∗ et = σxw ¯ β2 (xt − m1 ) − β(wt − m1 ) + υt for t > ξT  w x 2 σw The residual variance is again constructed in the usual fashion, yielding ) 2 ( ) σxw ( 2 σ2 ¯ 2 2 2 2 ¯ σe∗ = ξβ1 + (1 − ξ)β2 σx − 2 ξβ1 + (1 − ξ)β2 2 βσxw + xw β 2 σw + συ 4 σw σw ( 2 ) 2 σ2 ¯ σ2 ¯ 2 2 = ξβ1 + (1 − ξ)β2 σx − 2 xw β 2 + xw β 2 + συ 2 2 σw σw ( 2 ) 2 2 2 ¯ 2 = ξβ1 + (1 − ξ)β2 σx − ρ2 β 2 σx + συ xw ( ) ¯ = ξβ 2 + (1 − ξ)β 2 − ρ2 β 2 σ 2 + σ 2 ,

1 2 xw x υ



(43)



where ρxw is the correlation between xt and wt . Also in this case, we see that the variance of 2 the residuals consists of the original variance of the errors συ and an extra term related to the misspecified model with regard to xt . The increase with respect to the error variance is largest when the omitted factor is unrelated to other factors in the model, i.e., ρxw = 0. When there is some correlation, the factor wt can to some extend provide information on the omitted factor, and consequently the variance is reduced. When the correlation is perfectly positive or negative, the reduction is maximal, and Equation (43) reduces to Equation (35). Of course, in this particular situation, knowing wt implies knowing xt (up to a linear transformation) and the factor is not really omitted.



79



Finally we consider the test statistic. The asymptotic distribution of the estimators is quite similar to that in Equation (36)   σxw ¯  α + β1 m1 − 2 βm1   x w σw  ˆ   √  a1  σ ¯  ( ) −1 2 a2  − α + β2 m1 − xw βm1  → N 0, Σzz∗ σe∗ . T ˆ (44) x w  2  σw  c   σxw ¯ ˆ γ+ 2 β σw From this result, we derive the test statistic for the structural break test on the intercept, when there actually is a structural break in the omitted factor √ T ξ(1 − ξ)(β2 − β1 )m1 x . (45) χOFB = 2 σe∗

2 2 As we have established that σe∗ ≥ σe , we find χOFB ≤ χSFB . This means that omitting a factor to which the exposure exhibits a structural break actually reduces the bias in the test statistic. Consequently, investigating the effect of structural break in the factor exposure towards already included factors gives an upper bound to the effect that omitted factors with a structural break can have.



80



Publications in the Report Series Research in Management

ERIM Research Program: “Finance and Accounting”

2009 Sorting out Downside Beta Thierry Post, Pim van Vliet, and Simon Lansdorp ERS-2009-006-F&A http://hdl.handle.net/1765/14843 A Comment on: Storage and the Electricity Forward Premium Adriaan Bloys van Treslong and Ronald Huisman ERS-2009-042-F&A http://hdl.handle.net/1765/16246 Time Variation in Asset Return Dependence: Strength or Structure? Thijs Markwat, Erik Kole, and Dick van Dijk ERS-2009-052-F&A http://hdl.handle.net/1765/17096 Riding Bubbles Nadja Guenster, Erik Kole, and Ben Jacobsen ERS-2009-058-F&A http://hdl.handle.net/1765/17525







A complete overview of the ERIM Report Series Research in Management: https://ep.eur.nl/handle/1765/1 ERIM Research Programs: LIS Business Processes, Logistics and Information Systems ORG Organizing for Performance MKT Marketing F&A Finance and Accounting STR Strategy and Entrepreneurship